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4
Mathematics and Scientific Inquiry
Misha is only 3. You have to be 4 to join our club.
Im squooshing one raisin into every marshmallow. Theyre beanbag
chairs for bugs.
Add more water to the other glass to make them even.
First I did the napkins. Then I put spoons on top. Last the
plates with pop-up pebble pies.
I wont be 5 until after Jos. But then it will be my turn to be
the oldest.
My new shoes cost my mom eleventy-eight-one dollars.
Im running faster. I bet I get to the top of the hill before
you!
When the big hand points down at the 6, Im going to clap my
hands for clean-up.
Big yellow rings in this box and small ones there. Same for the
big and little green squares.
You be the daddy kitty because youre the tallest. Shes the baby
because shes the littlest. And Ill be the mommy so I can sit in the
middle.
Q uite naturally, and without recognizing them as such, young
children develop ideas about mathematics in the course of
their day-to-day lives. These childrens remarks, for example,
overheard in a preschool classroom, reflect an interest in
mathematical subjects that matter to themage, speed, time, size,
order. Summarizing the early mathematics knowledge base, Baroody
(2000) notes:
Researchers have accumulated a wealth of evidence that children
between the ages of 3 and 5 years of age actively construct a
variety of fundamentally important informal mathematical concepts
and strategies from their everyday experiences. Indeed, this
evidence indicates that they are predisposed, perhaps innately, to
attend to numerical situations and problems. (61)
Thus, in the past 25 years, studies of the development of early
mathematics have switched from looking at what children cannot do
to what they can do. Observations of children during free play, for
example, show them engaged in mathematical explorations and
applications, and sometimes these are surprisingly advanced
(Ginsburg, Inoue, & Seo 1999).
The typical early childhood curriculum, however, incorporates
little in the way of thoughtful and sustained early mathematics
experiences (Copley 2004). If mathematics is included, it tends
Mathematics and Scientific Inquiry 41
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to be limited to number, particularly counting. Yet young
children also spontaneously explore topics such as patterns,
shapes, and the transformations brought about by processes such as
adding and subtracting (Ginsburg, Inoue, & Seo 1999), and there
are foundational mathematical understandings children need to
develop in these areas.
Along with changing our ideas about what children can understand
has come rethinking of how to foster early mathematical
development.
Because young childrens experiences fundamentally shape their
attitude toward mathematics, an engaging and encouraging climate
for childrens early encounters with mathematics is important. It is
vital for young children to develop confidence in their ability to
understand and use mathematicsin other words, to see mathematics as
within their reach. (NAEYC & NCTM 2002)
Researchers and practitioners have developed different systems
for categorizing the mathematical areas in which young children
demonstrate interest and ability (e.g., Campbell 1999; Greenes
1999). In 2000 the National Council of Teachers of Mathematics
(NCTM) published Principles and Standards in School Mathematics,
which includes standards for grades preK2. And in 2002 NAEYC
published a joint position paper with NCTM supporting its standards
and offering recommendations for early mathematics education. The
NCTM standards are now widely cited in the field and used by many
state departments of education and local school districts to
develop comprehensive early mathematics curricula in preschool
programs and the primary grades.
The NCTM standards define five content areas: Number &
Operations, Geometry, Measurement, Algebra, and Data Analysis &
Probability. The standards are described in this chapter, along
with their application in the preschool years. NCTM also defines
five process standards, consistent with the strategies suggested
here: Problem Solving, Reasoning and Proof, Connections,
Communication, and Representation. Problem solving and reasoning,
as the position statement phrases it, are the heart of
mathematics:
While content represents the what of early childhood mathematics
education, the processes . . . make it possible for children to
acquire content knowledge. These processes develop over time and
when supported by well-designed opportunities to learn. Childrens
development and use of these processes are among the most
long-lasting and important achievements of mathematics education.
Experiences and intuitive ideas become truly mathematical as the
children reflect on them, represent them in various ways, and
connect them to other ideas. (NAEYC & NCTM 2002)
For further explanations of NCTMs process standards and
examples, see the work of Clements (2004) and Copley (2004), as
well as the National Council of Teachers of Mathematics Web site
(www.nctm.org).
Scientific inquiry and its relationship to mathematics Young
children are doing math when they measure and graph the daily
growth of bean seedlings, when they notice the changing patterns of
shadows on a wall, or when they predict how many more cups of sand
it will take to fill a hole and then check by counting. But they
are also doing science. Gelman and Brenneman point out that to do
science is to predict, test, measure, count, record, date ones
work, collaborate and communicate (2004, 156). Because of this
close connection between mathematics and science, I have chosen to
include the doing of sciencescientific inquiryin this chapter.
Science inquiry refers to the diverse ways in which scientists
study the natural world and propose explanations based on the
evidence derived from their work. Inquiry also refers to the
activities of students in which they develop knowledge and
understanding of scientific ideas, as well as an understanding of
how scientists study the natural world. (NCSESA 1996, 23)
Within science, scientific inquiry is perhaps the area that has
been most investigated because of young childrens evident interest
in observing and thinking about the world (Eshach & Fried
2005). Inquiry skills are in evidence when preschoolers are:
The Intentional Teacher 42
http:www.nctm.org
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> raising questions about objects and events around them;
> exploring objects, materials, and events by acting upon
them and noticing what happens;
> making careful observation of objects, organisms, and
events using all of their senses;
> describing, comparing, sorting, classifying, and ordering
in terms of observable characteristics and properties;
> using a variety of simple tools to extend their
observations;
> engaging in simple investigations in which they make
predictions, gather and interpret data, recognize simple patterns,
and draw conclusions;
> recording observations, explanations, and ideas through
multiple forms of representation including drawings, simple graphs,
writing, and movement;
> working collaboratively with others; and
> sharing and discussing ideas and listening to new
perspectives. (Worth & Grollman 2003, 18)
Doing inquiry and learning about inquiry are critical content
areas of science. Like any skills, they need to be learned and
practicedand this is where the other content areas of science make
their appearance. Teachers doing inquiry-based science choose
meaningful subject matter from the life, physical, earth, and space
sciences for young children to practice on. In other words,
Inquiry and subject matter are both important and cannot be
separated. Children may need direct instruction for some specific
skills, such as learning to use a magnifier. And they can practice
some skills on their own, such as categorization. But children
develop their inquiry skills as they investigate interesting
subject matter, and children build theories about interesting
subject matter through the use of inquiry skills. (Worth &
Grollman 2003, 156)
The connections between early mathematics and science are
reflected in NAEYCs own program accreditation criteria for science
(NAEYC 2005, Criterion 2.G) and technology (Criterion 2.H). The
NAEYC criteria note that children should have opportunities to
collect and represent data, make
inferences about what they observe, and have access to
technology. These opportunities and the associated learning are
also explicit in the guidelines for early mathematics education
(Criterion 2.F).
Young childrens development in mathematics and scientific
inquiry Young children, like those quoted at the beginning of the
chapter, start with only an intuitive or experiential understanding
of mathematics. They dont yet have the concepts or vocabulary they
need to use what they intuitively know or to connect their
knowledge to school mathematics. The preschool teachers task is to
find out what young children already understand and help them begin
to understand these things mathematically From ages 3 through 6,
children need many experiences that call on them to relate their
knowledge to the vocabulary and conceptual frameworks of
mathematicsin other words, to mathematize what they intuitively
grasp (NAEYC & NCTM 2002).
The goal of early mathematics education, then, is to build
mathematical power in young children (Baroody 2000). This power has
three components: a positive disposition to learning and using
mathematics; understanding and appreciating the importance of
mathematics; and engaging in the process of mathematical inquiry.
Turning childrens early and spontaneous mathematics play
(child-guided experience) into an awareness of mathematical
concepts and skills is at the heart of intentional teaching in this
area.
Similarly, in science we want to capitalize on childrens natural
inclination to learn about their world (Landry & Forman 1999,
133), expose them to the uses and benefits of scientific processes
in everyday life, and involve them in scientific inquiry as they
figure out how the world works. Here, too, children are naturally
inclined to explore their surroundings. But they depend on us to
give them a rich environment for inquiry and to develop their
child-guided discoveries into a growing understanding of how
science works.
Mathematics and Scientific Inquiry 43
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To promote childrens science inquiry, the intentional teacher
takes care to design the physical setting, plan the areas of
science children will focus on, and establish overall goals for
learning. Once children begin exploring, however, what actually
happens emerges from a dynamic interaction among the childrens
interests and questions, the
materials, and the teachers goals (Worth & Grollman 2003,
158).
In early mathematics and science, free exploration is
importantbut by itself it is not enough. There are concepts,
principles, and vocabulary that children will not construct on
their own. Even for those areas in which their investigations are
key,
Materials That Promote Mathematical and Scientific
Exploration
Children can manipulate, count, measure, and ask questions about
almost any object or kind of material. Yet there are some things
teachers should make sure to have in their classrooms to promote
exploration and thinking about the components of mathematics and
scientific inquiry.
Geometry and spatial sense
Number and operations
Printed items containing numbers and mathematical or scientific
symbolse.g., signs, labels, brochures, advertisements with charts
and graphs Things with numbers on theme.g., calculators, playing
cards, thermometers, simple board games with dice or spinners
Numbers made of wood, plastic, or cardboard (make sure they are
sturdy so children can hold, sort, copy, and trace them)
Discrete items children can easily counte.g., beads, blocks,
shells, poker chips, bottle caps Paired items to create one-to-one
correspondence e.g., pegs and pegboards, colored markers and tops,
egg cartons and plastic eggs
Materials and tools for filling and emptying water, sand;
scoops, shovels Everyday things to fit together and take apart
e.g., Legos, Tinkertoys, puzzles, boxes and lids, clothing with
different types of fasteners Attribute blocks that vary in shape,
size, color, thickness Tangram pieces Wooden and sturdy cardboard
blocks in conventional and unconventional shapes Containers and
covers in different shapes and sizes Materials to create
two-dimensional shapes e.g., string, pipe cleaners, yarn
Moldable materials to create three-dimensional shapese.g., clay,
dough, sand, beeswax Things with moving partse.g., kitchen
utensils, musical instruments, cameras Books that feature shapes
and locations, with illustrations from different perspectives
Photos of classroom materials and activities from different
viewpoints Materials that change with manipulation or time e.g.,
clay, play dough, computer drawing programs, sand, water, plants,
animals Materials to explore spatial concepts (over/under, up/down)
and to view things from different heights and positione.g.,
climbing equipment, empty boxes (large cartons from appliances and
furniture), boards Maps and diagrams
The Intentional Teacher 44
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children do not always construct mathematical or scientific
meanings from them. Clements (2001) suggests teachers consider
whether childrens thinking is developing or stalled. When it is
developing, they can continue observing. When it is stalled, it is
important to intervene (Seo 2003, 31). In this way, adult-guided
experience supplements child-guided exploration.
Teaching and learning in mathematics and scientific inquiry
Young children need many opportunities to represent, reinvent,
quantify, generalize, and refine their experiential and intuitive
understandings that might be called premathematical or emerging
Measurement
Ordered sets of materials in different sizese.g., nesting
blocks, measuring spoons, pillows, paintbrushes, drums Ordered
labels so children can find materials and return them to their
storage placee.g., tracings of measuring spoons in four sizes on
the pegboard in the house center Storage containers in graduated
sizes Materials that signal stopping and startinge.g., timers,
musical instruments, tape recorders
Materials that can be set to move at different rates of
speede.g., metronomes, wind-up toys Things in nature that move or
change at different ratese.g., slow- and fast-germinating seeds,
insects that creep and scurry Unconventional measuring toolse.g.,
yarn, ribbon, blocks, cubes, timers, ice cubes, containers of all
shapes and sizes Conventional measuring toolse.g., tape measures,
scales, clocks, grid paper, thermometers, measuring spoons,
graduated cylinders
Patterns, functions, and algebra
Materials with visual patternse.g., toys in bright colors and
black-and-white, dress-up clothes, curtains, upholstery Materials
to copy and create series and patterns e.g., beads, sticks, small
blocks, pegs and peg boards, writing and collage materials Shells
and other patterned items from nature Original artwork and
reproductions featuring patternse.g., weavings, baskets
Pattern blocks Routines that follow patterns Stories, poems, and
chants with repeated words and rhythms Songs with repetitions in
melody, rhythm, and words Computer programs that allow children to
recognize and create series and patterns
Data analysis
Tools for recording datae.g., clipboards, paper, Small objects
to represent counted quantities pencils, crayons, markers, chalk
e.g., buttons, acorns, pebbles Materials for diagramming or
graphing datae.g., Boxes and string for sorting and tying materials
newsprint pads and easels, graph paper with large into groups
grids, posterboard Sticky notes and masking tape for labeling
Mathematics and Scientific Inquiry 45
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mathematics. To do this effectively, intentional teachers design
programs so that children encounter concepts in depth and in a
logical sequence:
Because curriculum depth and coherence are important, unplanned
experiences with mathematics are clearly not enough. Effective
programs also include intentionally organized learning experiences
that build childrens understanding over time. Thus, early childhood
educators need to plan for childrens in-depth involvement with
mathematical ideas. . . . Depth is best achieved when the program
focuses on a number of key content areas rather than trying to
cover every topic or skill with equal weight. (NAEYC & NCTM
2002)
This need to focus on a limited number of key concepts and
skills at each level is further highlighted in NCTMs amendment of
its own standards, Curriculum Focal Points for Prekindergarten
through Grade 8 Mathematics (NCTM 2006). Intentional teachers use a
variety of approaches and strategies to achieve this focused
emphasis. They integrate mathematics into daily routines and across
other domains in the curriculum, but always in a coherent, planful
manner. This means that the mathematics experiences they include
follow logical sequences, allow depth and focus, and help children
move forward in knowledge and skills; it does not mean a grab bag
of experiences that seem to relate to a theme or project (NAEYC
& NCTM 2002).
In addition to integrating mathematics in childrens play,
classroom routines, and learning experiences in otherwise
nonmathematic parts of the curriculum, intentional teachers also
provide carefully planned experiences that focus childrens
attention on a particular mathematical idea:
Helping children name such ideas as horizontal or even and odd
as they find and create many examples of these categories provides
children with a means to connect and refer to their just-emerging
ideas. Such concepts can be introduced and explored in large and
small group activities and learning centers. Small groups are
particularly well suited to focusing childrens attention on an
idea. Moreover, in this setting the teacher is able to observe what
each child does and does not understand and engage each child
in the learning experience at his own level. (NAEYC & NCTM
2002)
Research points us to the materials and activities that foster
the development of mathematical and scientific concepts. Young
children are concrete, hands-on learners. They need to manipulate
materi-
The Intentional Teacher 46
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als to construct ideas about the physical properties of objects
and their transformation. Spontaneous investigations are most
common with discrete play objects such as Legos, blocks, or puzzles
and continuous materials such as sand, water, or clay (Ginsburg,
Inoue, & Seo 1999). Children tend to use mathematical and
scientific inquiry most frequently
during construction or pattern-making activities. Computers can
also play a role in early mathematics and science education, if the
technology is used appropriately (Hyson 2003). (See the box
Computer Technology.)
Perhaps less expected is the finding that mathematical and
scientific thinking is fostered by social
Mathematics and Scientific Inquiry 47
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interaction. When students share hypotheses and interpretations,
question one another, and are challenged to justify their
conclusions, they are more likely to correct their own thinking
(Campbell 1999). In fact, agreements and disagreements during
peer-to-peer dialogue more often prompt reflection and
reconsideration than does adult-delivered instruction (Baroody
2000), perhaps because many teachers underestimate childrens early
grasp of mathematical principles (Kamii 2000).
For that reason, understanding how children learn in the areas
of early mathematics and scientific thinking is essential to
meaningful teaching in this area. Research points to the
effectiveness of the following general support strategies:
> Encourage exploration and manipulation Provide materials
with diverse sensory attributes and allow children sufficient time
and space to discover their properties. At the same time, the ways
children use objects are often very different from those that we
intend or define (Seo 2003, 30). Teachers might not see art
materials or dramatic play props as mathematical manipulatives, but
children do, as they count the Velcro strips on a smock or the
rooms in a dollhouse. Indeed, there is nothing they do not count!
Its the same for science: The classroom pet may be there mostly for
the purpose of promoting social responsibility, but children are
also observing Sniffy the guinea pigs habits and life cycle.
> Observe and listenAttend to the questions children ask. The
problems they pose for themselves or to adults offer a window into
their mathematical and scientific thinking.
> Model, challenge, and coachDemonstrate hands-on activities
that children can imitate and modify. Provide experiences that
stretch their thinking. Discuss what does (not) work, pose
questions, and suggest alternative approaches to finding a
solution.
> Encourage reflection and self-correctionWhen children are
stuck or arrive at an incorrect mathematical solution or scientific
explanation, do not jump in to solve the problem or correct their
reasoning. Instead, provide hints to help children recon
sider their answers and figure out solutions or alternative
explanations on their own.
> Provide the language for mathematic and scientific
properties, processes, and relationships Introduce the language for
children to label their observations, describe transformations, and
share the reasoning behind their conclusions.
> Play games with mathematical elements Games invented for or
by children offer many opportunities, for example, to address
issues of (non)equivalence, spatial and temporal relations, and
measurement.
> Introduce mathematical and scientific content Children
enjoy good books about counting, especially when these invite
participation. Storybooks and nonfiction texts are also a wonderful
way to introduce real-life problems whose solutions depend on
mathematical reasoning. (See Resources for a Web site that suggests
books for young children related to early mathematics and science,
as well as other early learning topics.) Similarly, children are
fascinated by observing the natural world things that float or
sink, shifting cloud patterns, plant and animal habits. Thus
science should be considered content for mathematics and literacy
experiences (Gelman & Brenneman 2004, 156).
> Encourage peer interactionAs noted above, children can
sometimes explain mathematical and scientific ideas to their peers
more effectively than adults can. Sharing ideas, particularly
conflicting ones, prompts children to articulate and, where
necessary, modify their understanding.
In general, an investigative approach works better than a purely
didactic one. Begin with a worthwhile task, one that is
interesting, often complex, and creates a real need to learn or
practice. Experiencing mathematics in context is not only more
interesting to children but more meaningful (Baroody 2000, 64). It
also makes learning in both mathematics and science more likely and
more lasting. As Ginsburg and his colleagues note,
[Rote instruction that does not emphasize understanding] does
little to inculcate the spirit of mathematics learning to reason,
detect patterns, make conjectures, and perceive the beauty in
irregularitiesand may
The Intentional Teacher 48
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instead result in teaching children to dislike mathematics at an
earlier age than usual. Clearly the early childhood education
community should not implement at the preschool and kindergarten
levels the kinds of activities that the National Council of
Teachers of Mathematics is trying to eliminate in elementary
school! (Ginsburg, Inoue, & Seo 1999, 88)
Fitting the learning experience to the learning objective The
rest of this chapter describes what preschoolers learn as they
begin to acquire mathematical literacy across NCTMs five content
areas: Number & Operations, Geometry, Measurement, Algebra, and
Data Analysis & Probability. Some of NCTMs standards, however,
have labels that seem too sophisticated for what happens in
preschool mathematics. That is, are young children doing what older
children and adults know as geometry, algebra, or probability in
preschool? In writing its standards document, NCTM opted to use one
label for each area across the entire age range, from preK to grade
12, for a purpose. It wanted to emphasize that for each content
standard, children at every age are learning aspects of math that
relate to that standard. In this chapter, we have modified the
labels slightly to be more descriptive of the preschool learnings:
hence, Geometry and spatial sense and Patterns, functions, and
algebra and simply Data analysis.
Of those five areas, number and operations, geometry and spatial
sense, and measurement are areas particularly important for 3- to
6-year olds, because they help build young childrens foundation for
mathematics learning:
For this reason, researchers recommend that algebraic thinking
and data analysis/probability receive somewhat less emphasis in the
early years. The beginnings of ideas in these two areas, however,
should be woven into the curriculum where they fit most naturally
and seem most likely to promote understanding of the other topic
areas. Within this second tier of content areas, patterning (a
component of algebra) merits special mention because it is
acces
sible and interesting to young children, grows to undergird all
algebraic thinking, and supports the development of number, spatial
sense, and other conceptual areas. (NAEYC & NCTM 2002)
This chapter describes concepts and skills in the three key
areas, as well as in patterns, functions, and algebra and data
analysis. Each section is divided into those concepts and skills
that seem most likely to be learned, or best learned, through
childrens own explorations and discoveries (childguided experience)
versus those concepts and skills in which adult-guided experience
seems to be important in going beyond, as well as contributing to,
what children learn through their independent efforts. As with
every other domain, of course, this division is not rigid.
Number and operations This is the first of three areas that NCTM
(2000; 2006) has identified as being particularly important for
preschoolers. In the preschool years, number and operations focuses
on six elements, or goals for early learning. (For further
explanations and examples, see Clements 2004 and Copley 2004.)
Counting involves learning the sequence of number words,
identifying the quantity of items in a collection (knowing that the
last counting word tells how many), and recognizing counting
patterns (such as 21-22, 31-32, 41-42 . . .). Comparing and
ordering is determining which of two groups has more or less of
some attribute (comparing them according to which has the greater
or lesser quantity, size, age, or sweetness, for example), and
seriating, or ordering, objects according to some attribute
(length, color intensity, loudness). Composing and decomposing are
complementary: Composing is mentally or physically putting small
groups of objects together (e.g., two plus three blocks makes five
blocks), while decomposing is breaking a group into two or more
parts (e.g., five spoons is two spoons plus two spoons plus one
spoon).
Adding to and taking away is knowing that adding to a collection
makes it larger and subtracting makes it smaller. When this
understanding is combined with counting and (de)composing,
children
Mathematics and Scientific Inquiry 49
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can solve simple problems with increasing efficiency. Grouping
and place value are related: Creating sets of objects so each set
has the same quantity creates groups. Grouping in sets of 10 is the
basis for understanding place value later (i.e., making groups of
10 and then counting the leftovers). Equal partitioning is dividing
a collection into equal parts, a prerequisite to childrens
understanding of division and fractions.
To develop the mathematical understanding and skills encompassed
in these six areas, preschoolers need an optimal blend of
child-guided and adult-guided experiences. Because early
mathematical development depends so much on manipulating objects,
it is important that young children have ample opportunities to
work with materials that lend themselves to ordering, grouping and
regrouping, and so on. Children intuit certain properties and
processes from their spontaneous explorations, while adults help
them explore these ideas and provide the mathematical vocabulary to
describe the numerical properties and transformations they observe.
Adults also challenge children to try additional transformations
and to reflect on the results. These experiences and the role of
the intentional teacher are described below.
Of the key knowledge and skills in the area of number and
operations, intuiting number and its properties, as well as
performing informal arithmetic seem to develop best with
child-guided experience. On the other hand, adult-guided experience
seems to prove helpful for counting and numeration, as well as for
performing simple arithmetic.
C Child-guided experience is especially important for learnings
such as: C1 Intuiting number and its properties Even before they
learn how to count, young children come to an informal
understanding of quantity and equivalence. For example, they can
identify small quantities (up to four or five) by eyeballing them.
They use one-to-one correspondence to establish equivalence (e.g.,
matching each blue bead with a yellow one to see that there are
equal quantities of each). And they can make equal sets (i.e.,
make
groups) by putting one in each pile, then another in each pile,
and so on (e.g., to distribute an equal number of pretzels to each
person at the table). Although lacking a formal knowledge of sets
in a strict mathematical sense (defined as a collection of distinct
elements, such as a set of squares versus triangles), young
children can create groups and recognize when items share all or
some attributes with other items in a group.
Mathematicians, researchers, and practitioners agree that a
central objective of early mathematics education is developing
childrens number sense an intuition about numbers and their
magnitude, their relationship to real quantities, and the kinds of
operations that can be performed on them. Early number sense
includes this eyeballing ability, called subitizing (recognizing
quantities by sight alone, usually for quantities of four or fewer)
and establishing one-to-one correspondence, which is the foundation
of counting (i.e., linking a single number name with one, and only
one, object).
An ability to identify equivalence is also fundamental to
understanding number. Most 3-year-olds can recognize equivalence
between collections of one to four objects (e.g., two hearts and
two squares) without actually counting items. They can also
recognize equal collections in different arrangements as being the
same (e.g., three squares on the top and two on the bottom has the
same number as one square on top and four on the bottom). Most
4year-olds can make auditory-visual matches, such as equating the
sound of three dings with the sight of three dots. These findings
suggest that by age 3, children have already developed a nonverbal
representation of number, although its unclear what this mental
representation is like or how accurate it is. Regardless, they can
clearly represent and compare objects even before they can count
them (Baroody 2000).
The part-part-whole concept is the understanding that a whole
number (e.g., 7) can be represented as being made up of parts
(e.g., 4 and 3, or 5 and 2, or 6 and 1). Part-part-whole
representation is a precursor to number operations, helping
children understand addition and subtraction; most 3- and
4-year-olds can describe the parts of whole numbers
The Intentional Teacher 50
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up to 5, with understanding of larger numbers developing around
age 6 (Copley 2000, 5859).
Teaching strategies. Intuition develops with experience.
Teachers help young children develop their number sense by
surrounding them with a number-rich environment offering many
opportunities to work with materials and processes that rely on
numbers and their operations, as shown in the following
examples:
Display materials around the classroom printed with numerals and
mathematical or scientific symbols. Make sure the numerals are
large enough for children to see and are placed at their eye level.
Include manipulatives in the shape of numerals made of wood or
cardboard, as well as toys and other items with numerals on
them.
Offer materials and games that convey the concept of number,
such as dominoes and dice. Encourage children to explore them and
to find matches; for example, Can you find another domino with the
same number of dots?
Label and describe number phenomena that occur naturally in the
childrens play; for example, There are four wheels on Katies truck
and two more, or six, on Donalds and You found the second mitten
for your other hand.
Provide materials that allow children to explore one-to-one
correspondence, such as nuts/bolts and cups/saucers. Children will
also make one-to-one correspondences with any sets of materials
they are playing with; for example, giving each bear a plate or
ball.
Include materials that can be broken down and divided into
smaller parts, such as a lump of clay that can be divided into
smaller balls or a piece of fruit that can be sliced or separated
into sections. Unit blocks, Legos, and other toys with equal-size
parts that children can build up and then break down into
components also work well.
Offer materials that are the same in some ways but different in
others; for example, blocks of the same shape in different colors.
When children use these materials, make observations that
highlight
their attributes; for example, All the blocks in your tower are
square, but only some blocks are red.
C2 Performing informal arithmetic Informal arithmetic is
something similar to adding and subtracting nonquantitativelythat
is, without using numbers or other written symbols. Even before
receiving formal instruction, preschoolers often are able to solve
simple nonverbal addition and subtraction problems (e.g., two
children are drawing at the table when a third child sits down; one
child fetches another piece of paper for her). Children begin by
acting out problems with objects (setting out two blocks, and then
adding on one more). Later they can substitute representations
(e.g., tally marks on paper) for the physical objects and form
mental representations (visualizing two blocks, then adding one
more). Teachers can be helpful in encouraging representation (NCTM
2000).
Forming mental representations is significant: They understand
the most basic concept of additionit is a transformation that makes
a collection larger. Similarly, they understand the most basic
concept of subtractionit is a transformation that makes a
collection smaller (Baroody 2000, 63). Preschoolers can attain this
basic understanding of operations on their own, especially when
adults support its development. The understanding is fundamental to
later success in school mathematics.
In kindergarten, children sometimes solve simple multiplication
(grouping) and division (partitioning) problems by direct modeling
with objects. For example, in the problem, Jane has 10 pennies and
wants to give 2 to each friend. How many friends can she give
pennies to? the child would make piles with 2 pennies each and
count the number of piles to arrive at five friends. If the problem
were stated as Jane has 10 pennies and wants to give the same
number to each of five friends. How many pennies will each friend
get? the child would put a penny in each of five piles and repeat
the process until the pennies ran out, then count the number in
each pile to arrive at 2 pennies per friend. With adult guidance,
these
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informal strategies are replaced by formal number knowledge and
counting strategies (Campbell 1999).
Teaching strategies. Because preschoolers tend to think
concretely, handling objects and working with visual
representations help them carry out and understand operations.
Below are examples of strategies teachers can use to promote this
learning:
Provide many small items that children can group and regroup,
adding and subtracting units.
Pose simple addition and subtraction problems in the course of
everyday experiences. For example, after a child sets the table,
say Remember that Thomas is out sick today or Mrs. King is going to
join us for snack and see whether they subtract or add a place
setting. Or during block time, say Jane wants to make her wall one
row higher. How many more blocks will she need?
Pose simple multiplication or division problems that children
can solve using concrete objects. For example, give a child a
collection of objects at small group and say Give the same number
to everyone. Or say, There are five children and everyone wants two
scarves to wave in the wind. How many scarves will we need to bring
outside?
A Adult-guided experience is especially important for learnings
such as: A1 Counting and numeration For young children, counting
and numeration (reading, writing, and naming numbers) involves
understanding numbers, which is knowing the number names and the
position of each one in the sequence, ordinal numbers (e.g., first,
second, . . .), and cardinal numbers (one, two, . . .); notation,
which is reading and writing numerals and recognizing the simple
mathematical symbols +, , and = ; counting, which is determining
quantity and equivalence; and sets, which involves creating and
labeling collections and understanding all and some. These are each
elaborated below.
As with learning letter names and shapes, children cannot
acquire knowledge of number
names and numerals unless adults give them this information. At
times, children will ask, How do you write three? or What comes
after 10? but the intentional teacher also is proactive in
introducing the vocabulary and symbols children need to understand
and represent mathematical ideas (Campbell 1999). With adult
guidance, children can then apply this knowledge to solve problems,
including those of measurement and data analysis.
Early counting is finding out how many, which is a powerful
problem-solver and essential to comparing quantities. Research
(Gelman & Gallistel 1978) has identified five principles of
counting: (1) stable order (2 always follows 1); (2) one-to-one
correspondence (each object is assigned a unique counting name);
(3) cardinality (the last counting name identifies how many); (4)
order irrelevance (objects can be counted in any order without
changing the quantity); and (5) abstraction (any set of objects can
be counted). Adult-guided experience helps preschoolers develop
these understandings.
Older preschoolers use counting to determine that two sets of
objects are equivalent. Between the ages of 3 and 4, as they
acquire verbal counting skills, children gain a tool more powerful
than their earlier subitizing for representing and comparing
numbers, including collections larger than four items. They
recognize the same number name principle (two collections are equal
if they share the same number name, despite any differences in
physical appearance). Children generalize this principle to any
size collection they can count. Similarly, by counting and
comparing two unequal collections, preschoolers can discover the
larger number principle (the later a number word appears in the
sequence, the larger it is). By age 4, many preschool children can
name and count numbers up to 10 and compare numbers up to 5. When
they have ample opportunities to learn the counting sequence,
children often learn to name and count to 20 by age 5 (Clements
2004). They are also fascinated by large numbers, such as 100 or a
gazillion, even if they only know them as number names without a
true sense of their value.
Equal partitioning builds on and is related to the concept of
equivalence. Equal partitioning is the
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process of dividing something (e.g., a plate of eight cookies)
into equal-size parts (e.g., to serve four children). Children as
young as 4 or 5 begin to solve such problems concretely, using
strategies such as dividing the objects into the requisite number
of piles (four) and then counting how many are in each pile
(Baroody 2000).
Teaching strategies. Psychologist Howard Gardner says,
Preschoolers see the world as an arena for counting. Children want
to count everything (1991, 75). Being creative, teachers can invent
or take advantage of many situations to count objects and events in
childrens daily lives. For example:
Notice things children typically compare (the number of blocks
in a tower, their ages), and provide materials and experiences
based on these observations. Think of fun and unusual things to
count; for example, the number of mosquito bites on an ankle.
Make numerals prominent. Place numerals of different materials,
sizes, and colors throughout the classroom. Provide cards with dots
and numerals for children to explore, sort, and arrange in order.
Use numerals on sign-up sheets so children can indicate not only
the order but also how many turns they want or for how long (two
minutes, three flips of the sand-timer). Children can indicate
their preferences with numerals or other marks (stars, checks, hash
marks).
Use written numerals and encourage children to write them. For
example, when they play store, encourage them to write size and
price labels, orders, and the amount of the bill.
Use everyday activities for number learning and practice. For
example, as children gather or distribute countable materials,
engage them in counting at clean-up (counting items as theyre
collected and put away), small group (handing out one glue bottle
per child), and choice time (distributing playing cards). At snack
or mealtimes, ask the table setter to count children to determine
how many place settings are needed. Pose simple number problems
such as Our group has six, but Celia is sick today. How many
napkins will we need? or
How many cups of sand will it take to fill the hole?
Use games as a natural yet structured way to develop counting
skills. Examples include board games with dice (moving a piece the
corresponding number of places) or physical movement challenges
(counting the number of times the tossed beanbag lands in the
bowl).
Use childrens own questions as the springboard for teachable
moments. For example, Baroody (2000) imagines an incident when
Diane says to her teacher,
My birthday is next week, how old will I be? Will I be older
than Barbara? The teacher could simply answer, Youll be 4, but
Barbara is 5 so shes still older. Or, the teacher can respond by
saying, Class, Diane has some interesting questions with which she
needs help. If she is 3 years old now, how can she figure out how
old shell be on her next birthday? The teacher could follow up by
posing a problem involving both number-after and number-comparison
skills: If Barbara is 5 years old and Diane is 4 years old, how
could we figure out who is older? (65)
Use childrens literature. Not only are there many appealing
counting books, but there are storybooks in which mathematics is
used to solve a problem. For example, read books where the story is
about sharing a quantity of something fairly. Before the problem is
solved in the book, ask children to suggest solutions by trying
them out with materials or working through simple ideas in their
heads. Children can work alone or in pairs. After reading the book,
encourage them to comment on the solution(s) in the text. As a
follow-up, they might role-play the same or similar situations
using props you supply or they make themselves.
A2 Performing simple arithmetic Younger preschoolers perform
simple arithmetic qualitatively. Older preschoolers, however, begin
to add and subtract whole numbers quantitatively that is, using
numerals to abstractly represent numbers of objects, rather than
physically manipulating or visualizing the objects. They are able
to do this because they can hold a representation of quantities in
their minds. For example, they may say out loud, Two and one more
is three or If Kenny isnt here
Mathematics and Scientific Inquiry 53
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today, I only need four napkins. Although they can do this most
readily with numerals up to 5, some preschoolers can handle numbers
up to 10.
Research shows they may also be capable of adding and
subtracting very simple fractions. For example, when researchers
hid part of a circle behind a screen and then hid another fraction,
children could visually identify what the total amount was. They
understood that two halves made a whole, a half plus a quarter
circle resulted in a three-quarters circle, and so on (Mix, Levine,
& Huttenlocher 1999). Such research suggests that children can
grasp the basic idea behind simple fractions if adults pose
interesting challenges.
Teaching strategies. Arithmetic follows fixed rules or
conventions. Like combining letters into words, performing
operations on numbers depends on knowing these rules. With support
from their teachers, preschoolers are capable of solving simple
arithmetic problems that come up in play and exploration. They are
also motivated to use arithmetic like grown-ups. Teachers can
therefore readily implement strategies such as the following to
enhance young childrens early understanding and use of
arithmetic:
Use real objects when helping children work through arithmetic
problems. For example, if a child is building a tower of three
blocks, count them with the child, and ask how many blocks there
would be if the child added two more to make it taller. Wonder
aloud how many blocks would be left if the child made it three
blocks shorter. The child can add or subtract the actual blocks and
count the result to determine the answer.
Pose challenges that build on childrens interests. For example,
if a child has drawn a picture of a dog, wonder aloud whether the
child can draw a dog twice as big or half as big.
Encourage children to use arithmetic to answer their own
questions. For example, if a child says, My daddy wants to know how
many cupcakes to bring for my birthday tomorrow, you could reply
Well, there are 16 children and two teachers. Plus your daddy, and
your brother will be here, too.
How can we figure out how many cupcakes youll need to bring?
Encourage children to reflect on their arithmetic solutions
rather than telling them if theyre right or wrong. When children
are stumped (though not yet frustrated) or arrive at erroneous
answers, resist the temptation to give the answer or correct them.
Instead, offer comments or pose questions that encourage them to
rethink their solutions. Baroody (2000) gives this example:
Kamie concluded that 5 and two more must be 6. Instead of
telling the girl she was wrong and that the correct sum was 7, her
teacher asked, How much do you think 5 and one more is? After Kamie
concluded it was 6, she set about recalculating 5 and two more.
Apparently, she realized that both 5 and one more and 5 and two
more could not have the same answer. The teachers question prompted
her to reconsider her first answer. (66)
Start with one fraction at a time. For example, children are
fascinated by the concept of one half. If they learnreally
learnthrough repeated experiences that half means two parts are the
same and together they make up a whole, then they can generalize
this concept later to thirds, quarters, and so on.
Geometry and spatial sense This is the second of three areas
NCTM (2000; 2006) has identified as being particularly important
for preschoolers. In the preschool years, learning about geometry
and spatial sense focuses on four elements: Shape refers to the
outline or contour (form) of objects and comprises identifying two-
and three-dimensional shapes. Locations, directions, and
coordinates refers to understanding the relationship of objects in
the environment. Transformation and symmetry is the process of
moving (sliding, rotating, flipping) shapes to determine whether
they are the same. It also involves building larger shapes from
smaller shapes, a common construction activity in preschool.
Visualization and spatial reasoning is creating mental images of
geometric objects, examining them, and transforming them. At first
childrens mental representations are static; that is, children
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cannot manipulate them. Later children can move and transform
images mentally; for example, deciding whether a chair will fit in
a given space or imaging a puzzle piece rotated.
Spatial concepts and language are closely related; for example,
where someone stands determines whether he is in front of or behind
another object. Thus, it is important that young children be given
numerous opportunities to develop their spatial and language
abilities in tandem (Greenes 1999, 42). Because society has
specific conventions for labeling various shapes, transformations,
and especially concepts of position, location, and so on, teachers
especially need to enhance childrens descriptive vocabulary in this
domain.
Of the key knowledge and skills in the area of geometry and
spatial sense, child-guided experience seems most helpful for
creating familiarity with two- and three-dimensional shapes and
their attributes, as well as for orienting self and objects in
space. To create, name, and transform shapes, on the other hand, as
well as to articulate position, location, direction, and distance,
adult-guided experience seems necessary.
C Child-guided experience is especially important for learnings
such as: C1 Familiarity with two- and three-dimensional shapes and
their attributes For young children, shape knowledge is a
combination of visual and tactile exploration, which begins in
infancy. During the preprimary years, NCTM expects children to
recognize, name, build, draw, compare, and sort two- and
three-dimensional shapes. Although most adults support childrens
recognition of two-dimensional shapes, they often overlook the need
to give children experiences with three-dimensional shapes, which
focus their attention on geometrical features. For example,
exploring the rolling of cylinders and other shapes helps children
to understand the properties of the circle versus the ellipse.
These skills involve perceiving (differentiating) such attributes
as lines and cubes;
circles, cylinders, and globes; sides and edges; corners,
angles, and so on. Preschoolers are also engaged in investigating
transformations with shapes (composing and decomposing), and they
demonstrate an intuitive understanding of symmetry. (Note: Children
need adult-guided experiences to learn to accurately label and
describe transformations and symmetry.)
Teaching strategies. Communication skills are important in all
areas of mathematics, but especially so in geometry. Spatial
concepts and language are closely relatedwords facilitate an
understanding of such concepts as on top of, next to, behind, and
inside. For example:
Introduce both two- and three-dimensional shapes, giving
children opportunities to explore them. Include both regular and
irregular shapes. Engage children in drawing and tracing the
shapes. Provide models (drawings, molds, maquettes) and tools
children can use to trace or copy them. Visual and physical shapes
help young children grasp the essential attributes of each.
Encourage children to sort shapes and provide reasons for their
groupings. Encourage them to describe why objects are not
alike.
Encourage children to combine (compose) and take apart
(decompose) shapes to create new shapes; for example, combining two
triangles to make a square or rectangle (composing), and vice versa
(decomposing). Engage them in discussions about these
transformations.
Provide materials that have vertical (i.e., left/ right halves
are identical) or horizontal (i.e., top/ bottom halves are
identical) symmetry; for example, doll clothes, a teeter-totter,
and a toy airplane. For contrast, provide similar but asymmetric
materials; for example, a glove, slide, and toy crane. Engage
children in discussing how the two sides (or top and bottom) of
objects are the same (symmetrical) or different (asymmetrical).
C2 Orienting self and objects in space Spatial relationshow
objects are oriented in space and in relation to one anotherare the
foundation
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of geometry, which involves understanding and working with the
relationships of points, lines, angles, surfaces, and solids.
Compared with toddlers, preschoolers navigate their bodies and move
objects with greater skill and confidence. Younger children still
tend to see and describe space from their own perspective
(egocentrism), but older preschoolers can begin to represent and
describe things from another persons point of view (perspective
taking).
Teaching strategies. Because mathematics is the search for
relationships, early instruction should focus on physical
experiences through which children construct understandings about
space. Teachers do this primarily by providing materials and
allowing children ample time to explore them:
Create different types of space in the classroom and outdoor
areasmall spaces for children to maneuver into and around; large
open areas where children can move about freely; spaces to crawl
over and under, in and out, up and down, and around and through.
Ask and talk with children about their relationships with objects
and with one another.
Provide materials, time, and ample space to build with
construction toys. For example, notice all the relative dimension
and position concepts Trey and his friends used when they made a
bus with large wooden blocks and invited their classmates and the
teacher to get onboard:
The group quickly decided the bus was too small, so they made it
bigger by adding many more seats. The children worked hard fitting
the big wooden blocks end-to-end to make the bus longer. They made
a drivers seat up front and made a steering wheel to fit on top of
the dashboard. They also decided to build a refrigerator in the
back of the pretend bus. Trey said it needed to be on the back
wall, but in the middle of the aisle. (Tompkins 1996b, 221)
Provide other materials to move and rearrange; for example, doll
house furniture or pedestals to display artwork. Provide materials
children can use to organize and construct collages.
A Adult-guided experience is especially important for learnings
such as: A1 Creating, naming, and transforming shapes The ability
to accurately name, describe, and compare shape, size (scale), and
volume is important for children to acquire during the preschool
years. With appropriate experiences and input they learn to
transform shapes to achieve a desired result and describe the
transformation (Im making this bridge longer by adding more blocks
at the end and holding it up in the middle). They can also create
and label symmetry in their two- and three-dimensional creations.
Language is critical in all these activities. Therefore, as
vocabulary expands, so does geometric understanding.
Teaching strategies. Building on preschoolers explorations of
shapes, teachers should explicitly focus the childrens attention on
features and what the shapes will do (e.g., Which of these shapes
can roll?) and provide words for these characteristics. Children
should be given opportunities to identify shapes in various
transformations, including reflections and rotations and
(de)compositions. For example:
Comment and ask children about differences in the size and scale
of things that interest them; for example, their own bodies, food
portions, piles of blocks. Encourage them to alter two- and
three-dimensional materials and comment on the transformations,
including whether their manipulations resulted in regular or
irregular forms.
Identify and label shapes and their characteristics throughout
the childrens environment (classroom, school, community). Go on a
shape hunt in the classroom (e.g., a triangle search). Use
increasingly sophisticated vocabulary words; for example, say On
our walk, lets look for all the square signs or You used cubes and
rectangular blocks to build your dollhouse. Remember to supply
names of three-dimensional as well as two-dimensional shapes.
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Encourage the exploration of shapes beyond conventional ones
such as circles, squares, and triangles. Young children enjoy
hearing and learning names such as cylinder and trapezoid. Even if
they do not fully grasp the meaning and characteristics, they
become attuned to the variety of spatial phenomena in the world.
Also important is giving children diverse examples of triangles and
other shapes, not just the equilateral triangle that is the only
example offered in many classrooms.
Use printed materials to focus on shape. Cut out photographs
from magazines that feature shape pictures and encourage children
to sort them. Create a shape scrapbook for the book area. Encourage
children to build structures like those in story and information
books. Refer to the books and talk with children about their choice
of materials, how they match the attributes in the illustrations,
and how they are recreating or modifying the structures or
both.
Challenge children to imagine what their structures would look
like with one or more elements transformed, for example, in
location or orientation. Encourage them to represent and verify
their predictions. For example, in Building Structures with Young
Children, Chalufour and Worth share this note from a preschool
teacher:
I brought a whiteboard and markers over to the block area
because Abigail was having a hard time imagining what her tower
would look like if it were built with the blocks placed vertically
instead of horizontally, as she had done. Not only did it help her
to see a drawing of a tower built with verticals, but Adam came up
to the drawing and pointed to one of the blocks near the top of the
drawing, declaring that he didnt think it would balance on top of
the one under it. So he and Abigail proceeded to use the drawing to
build a tower and, lo and behold, Adam was right! Tomorrow Im going
to invite him to tell the group about the event. We can ask Adam
how he knew that the vertical wouldnt balance. (2004, 45)
A2 Articulating position, location, direction, and distance
Expectations in this area involve concepts of position and relative
position, direction and distance, and location (NCTM 2000). With
appropriate adult guidance, preschoolers can use position and
direc
tion words and follow orientation directions. They also are able
to begin moving beyond their egocentric perceptions to predict
anothers perspective. For example, with experience they can
describe how someone else would see something from his perspective,
and can give appropriate directions or instructions to another
person.
Teaching strategies. Teachers need to supply vocabulary, of
course. But preschoolers still master such ideas through a
combination of concrete experience and mental imagery, so teachers
need to provide many opportunities for them to represent these
concepts in two- and three-dimensional ways:
Make comments and ask questions that focus on location and
direction; for example, You attached the sides by putting a long
piece of string between the two shorter ones or Where will your
road turn when it reaches the wall? Comment on naturally occurring
position situations, such as Larry is climbing the steps to the
slide, Corys next, and last is Jessica.
Use various types of visual representations to focus on these
concepts. Engage children with making and interpreting mapsfor
finding a hidden object, for example. Children can draw diagrams of
the classroom, their rooms at home, and other familiar places. Ask
them about the placement of the objects. Comment on the location of
things in their drawings using position words; for example, You
have a big poster over your bed or Whats poking out behind the
curtain? Ask children to draw, paint, build, or use their bodies to
represent favorite books featuring characters or objects in
relation to one another. For example, ask them to draw the three
bears sitting around the table or to lie down next to one another
like the three bears in their beds. Because society has specific
conventions for labeling various concepts of position, location,
and so on, teachers are especially needed to enhance childrens
descriptive vocabulary in this domain.
Create occasions for children to give directions; for example,
when helping one another or leading during large group. This
requires them to use position and direction words such as Hold the
top and
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push down hard into the dough or Stretch your arms over your
head and then bend down to touch your toes. Encourage children to
volunteer as the leader.
Use movement to focus on spatial concepts. Provide objects that
can be thrown safely, such as beanbags and foam balls, and interact
with children about distance. Use simple movement directions for
games and dances at large group such as Hokey Pokey. Invent
variations to games and dances by frequently modeling the adding of
a new twist. Get the children engaged in making up variations of
their own.
Talk about trips children take with their families or about
walks and field trips with the class. For example, Does your
grandma live close to you or far away? or We took a long ride to
the zoo on the bus, but after we parked, it was just a short walk
to the monkey cage.
Measurement This is the third of three areas NCTM (2000; 2006)
has identified as being particularly important for preschoolers. In
the preschool years, learning about measurement focuses on two
elements: Attributes, units, and processes refers to developing
concepts about size and quantity, arranging objects to compare
them, estimating differences (e.g., by eyeballing, lifting), and
quantifying differences with nonstandard (e.g., footsteps) and
standard (e.g., tape measure) tools. Techniques and tools comprises
learning measuring rules such as starting at zero, aligning or
equalizing beginning points, and not allowing gaps. It also
includes becoming familiar with standard measuring tools such as
rulers, scales, stopwatches, and thermometers. As with spatial
concepts, measurement benefits from language, especially comparison
words.
Of the key knowledge and skills in the area of measurement,
comparing (seriating) or estimating without counting or measuring
seems to develop best with child-guided experience, while
adult-guided experience seems integral for counting or measuring to
quantify differences.
C Child-guided experience is especially important for learnings
such as: C1 Comparing (seriating) or estimating without counting or
measuring Young children are able to grasp the basic concept of one
thing being bigger, longer, heavier, and the like, relative to
another. Making comparisons is the beginning of measurement.
According to NCTMs standards (2000), preschoolers should be engaged
in comparing length, capacity, weight, area, volume, time, and
temperature.
At first, children make qualitative comparisons by matching or
ordering things (Stacy is the short one, and Bonnie is tall or My
cup holds more water than yours) rather than quantitative
comparisons that use counting or measuring. To estimate, they use
their various senses, such as eyeballing (visual), lifting
(kinesthetic), or listening (auditory). They may compare length by
aligning blocks on the bottom and seeing how much they stick out on
top, or listen to instruments to compare their loudness.
Teaching strategies. Teachers can draw on childrens interest in
comparing to focus their attention on quantitative and qualitative
differences. Examples abound in mathematical and scientific
applications, including those suggested here:
Make comments and ask questions using comparison words (Which of
these is longer? or Does everyone have the same number of cookies
now?) Ask children whether they think something is wider (softer,
heavier, louder, colder) than something else.
Provide ordered sets of materials in different sizes, such as
nesting blocks, measuring spoons, pillows, paintbrushes, and drums.
Affix ordered labels that children can use to find materials and
return them to their storage place; for example, four sizes of
measuring spoons traced on the peg board in the house center.
Provide storage containers in graduated sizes.
Encourage children to move at different rates throughout the day
and comment on relative speed. Make transitions fun by asking
children to proceed
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to another area or activity as slow as a snail or as fast as a
rocket. Acknowledge their observations about speed and what affects
it. A preschool teacher shared this anecdote:
At outside time, James was pushing two children around the
playground on the toy taxi. When the adult asked if she could have
a ride, he said Sure. After going around two more times, James
stopped the taxi and said, Get off. Youre too fat and I cant go
fast. Acknowledging the validity (if not the kindness) of his
observation, the teacher got off the taxi so he could move at a
faster clip. (Graves 1996, 208).
Call childrens attention to graduated changes in nature. Comment
on seasonal fluctuations in temperature (e.g., It feels colder now
than it did when we went to the pumpkin patch. Were wearing heavier
jackets.). Plant a garden and ask children how long they think it
will take before the seeds germinate, the vegetables are ready to
eat, and so on.
A Adult-guided experience is especially important for learnings
such as: A1 Counting or measuring to quantify differences Many
older preschoolers and kindergarten children are able to understand
the idea of standard units, and with well-conceived learning
experiences, they can begin to determine differences in quantity
by
systematic measurement. They use their knowledge of number to
make comparisons. At first, they use nonstandard units such as how
many steps it takes to cross the schoolyard in each direction or
the number of song verses to clean up different areas of the room.
With teachers assistance, they acquire the understanding that it is
useful to employ conventional units and measuring devices, such as
inches on a ruler or minutes on a clock.
Teaching strategies. There are many opportunities throughout the
day for children to engage in measurement; for example, when they
are building something or resolving a dispute. However, it usually
does not occur to preschoolers to measure or quantify things to
solve these problems. Adults can actively encourage children to use
measurement in these situations. For example:
Provide conventional and unconventional measuring devices, and
encourage children to use them to answer questions or solve
problems. Conventional devices include rulers, tape measures,
clocks, metronomes, kitchen timers, and spring and balance scales.
Unconventional ones include string or paper towel tubes for length,
sand timers for duration, grocery bags for volume (three bags of
blocks were needed to make a tall tower, only one bag to make a
short one), unmarked bags of clay or sand
Gathering Data to Resolve a Social Conflict Undertaking an
investigation with adult help is one Devon: Hey! Im bigger than
you. I get to go first! way to resolve disputes (for more, see
Chapter 5). John: No youre not. Im the biggest one. Some conflict
situations lend themselves to col- Liza: Lets measure and find out.
lecting information that can be quantified and interpreted to reach
a fair solution. In this example, The children stood against the
wall and asked their
teacher to make a chalk mark where each of their a group of
older preschoolers resolve a dispute by heads touched. Then they
got the tape measure and asked their teacher to write down how
many
measuring heights and are surprised by the result!
John, Liza, and Devon argued about the order of inches tall each
child was. She wrote 41 next to the turn-taking to drive the big
truck around the J, 42 next to the D, and 44 next to the
L.playground. They decided the biggest person should go first, then
the next biggest, then the Liza: Its me! Im the biggest! smallest.
They would make a list and check off Devon: Yeah, and Im next. John
is last. each name as that person finished a turn. John John: But I
can stay on the longest because theres chalked J, D, L on the
blacktop and moved to get no one after me! on the truck. But the
conflict was not yet over:
Mathematics and Scientific Inquiry 59
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for weight. Children can also develop their own devices. When
children ask measurement-related questions (Which is heavier?) or
have disputes (I am too taller by a whole, big lot!), ask them
which of these tools might help them arrive at an answer or
solution.
Pose measurement challenges that children will be motivated to
solve; for example, I wonder how many cups of sand it will take to
fill all 12 muffin tins? Ask How many more . . . ? questions, such
as How many more pieces of train track will you need to close the
circle? Heres an amusing challenge shared by a curriculum developer
and writer who works with preschoolers:
I was stretched out on the floor against a wall. I said, I
wonder how many mes long the wall is? The children thought this was
very funny, but they were intrigued to figure out the answer. Some
estimated by simply envisioning the response. Others wanted to use
the direct route, having me move and stretch out while they counted
the number of repetitions. When I said I was too comfortable and
didnt want to move, the children had to come up with another
solution. They decided that two of them equaled one of me, so they
stretched in a line along the wall and counted how many of them it
took. With my help, they then divided the number of children in
half. (Stuart Murphy, 2004, pers. comm.)
When resolving social conflicts with children, ask how they
could measure to guarantee a fair solution; for example, to make
sure everyone gets to play with a toy the same amount of time.
Use visual models to help children understand and quantify
differences. For example, make a daily routine chart where the
length of each part in inches is proportional to its duration in
minutes. Give time checks (Five minutes to clean-up) with visual
and auditory cues.
Create opportunities for group construction projects, such as
laying out a garden, making a bed for each doll within a defined
space, or recreating a supermarket after a class field trip. These
often lead to situations where children have different opinions and
need to measure to find out who is right or what solution will
work. Sometimes you will need to suggest this method of resolving
the difference of opinion.
Include units of measurement when sharing information with
children; for example, I went grocery shopping for an hour last
night or My puppy gained 5 pounds since the last time I took him to
the vet.
Patterns, functions, and algebra In the preschool years,
learning about patterns, functions, and algebra focuses on two
elements: Identifying patterns involves recognizing and copying
patterns and determining the core unit of a repeating pattern. It
includes visual, auditory, and movement patterns. Deciphering
patterns requires inductive reasoning, which is also a precursor to
understanding probability. Describing change is using language to
describe the state or status of something before and after a
transformation. For example, When I was a baby, I couldnt drink out
of a cup or When we raised the ramp a little higher, my car went
all the way to the book shelf.
Of the key knowledge and skills in the area of patterns,
functions, and algebra, child-guided experience seems to help
children recognize, copy, and create simple patterns and also
recognize naturally occurring change. For children to identify and
extend complex patterns and to control change, on the other hand,
children seem to benefit most from adult-guided experience.
C Child-guided experience is especially important for learnings
such as: C1 Recognizing, copying, and creating simple patterns For
young children, this area encompasses an awareness of patterns in
the environment (visual, auditory, temporal, movement).
Preschoolers can acquire the ability to copy or create simple
patterns with two elements, such as A-B-A-B or AA-BB. Even before
they know the word pattern, children notice recurring designs or
routines in their lives, whether it be on their clothing, the
stripes on a kittens back, or the order of each days
activities.
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Preschoolers generally need at least three repetitions of a
pattern before they can recognize or repeat it.
Teaching strategies. Patterns and series of objects or events
are plentiful in the world. Teachers can actively help children
become aware of common patterns and series. Simple observations and
questions can lead them to notice and create regularity and
repetition. For example:
Ask children to do or make things that involve series and
patterns. For example, at small group, give children drawing or
sculpting materials and invite them to represent their familiesfrom
the smallest to the biggest members. Other materials that lend
themselves to pattern making include string and beads in different
colors and shapes (e.g., to make a necklace), multi-colored blocks
in graduated sizes (e.g., to make a train), and pegs and pegboards
(e.g., to make a design).
Acknowledge the patterns children spontaneously create in art
and construction projects. When they are busy building, acknowledge
their work with a smile and a descriptive statement such as, I see
a pattern in your tower. First you used two rectangles, then you
used a cylinder, and then you added two more rectangles and a
cylinder or This reminds me of the Eiffel Tower. Its wide at the
bottom and becomes narrow at the top (Chalufour & Worth 2004,
38). Music also provides many opportunities for calling attention
to patterns; for example, You sounded two loud, one soft, two loud,
and one soft beat with the rhythm sticks. Movement provides another
source for constructing patterns; for example, a series of two or
three steps repeated in sequence (side, side, hop, side, side,
hop).
At large group, encourage children to move their bodies into
graduated positions such as lying, sitting, and standing. Move
through transitions at slow, medium, and fast paces.
Read and act out stories in which size, voice, or other
graduated qualities play a role, such as The Three Bears or The
Three Billy Goats Gruff. At small group, ask children to make beds
for the three bears
with play dough. At large group, have them choose which
instrument the papa, mama, or baby bear would play, depending on
variations in pitch or loudness.
C2 Recognizing naturally occurring change Noticing and
describing changes includes identifying what variable or variables
are causal. This is a mathematics concept, but it is also prominent
in science in childrens developing awareness of changes in the
world around them and possible reasons for these. For example,
children see changes in their own bodies (e.g., getting taller) or
the growth of a flower. Although they are often unable to identify
the causal factor accurately, young children do make tentative
guesses, both right and wrong, about the changes they see. For
example, Im 5 today. That means Im taller or The flower grew up
because the wind blew on it from the bottom.
Teaching strategies. The most important strategy teachers can
follow in this area is to notice and acknowledge childrens
awareness of changes in their environment and initiate situations
in which change can be created, observed, and investigated; for
example, by discussing the growth of vegetables in the school
garden or experimenting with color mixing at the easel. Repeating
and extending childrens comments about the changes they observe is
a signal you are listening to them. Calling their attention to
change and showing you are interested in their reaction, and in
their explanations, is also a form of acknowledgment. For
example:
Repeat childrens comments to acknowledge their spontaneous
seriation. When LaToya said, These giants are hungrier because they
have bigger teeth, her teacher agreed: Those bigger teeth will help
the monsters eat lots more food.
Extend childrens comments. Josh was washing his hands at the
sink when his teacher turned on the water in the next sink full
blast. Josh said Mine is running slow. She turned down her water
and said I made mine slower like yours.
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Call childrens attention to cycles in nature with concrete
examples. Point out the seasonal variations in schoolyard plants or
the changing thickness of childrens jackets from fall (lightweight)
to winter (heavy) to spring (back to light). Document changes with
photographs.
A Adult-guided experience is especially important for learnings
such as: A1 Identifying and extending complex patterns Simple
patterning is something young children do spontaneously. With
experience and adult input, they learn to do more. For example,
older preschoolers and kindergartners are able to analyze,
replicate, and extend the core unit of a complex repeating pattern
with three or more elements (A-BC-A-B-C; 1-22-3-22-1), provided
they see or hear it several times (Clements 2004). They can also
begin to recognize what are called growing patterns that is,
patterns where successive elements differ (rather than repeat) but
still proceed according to an underlying principle, such as
counting by ones or twos (2-4-6). The same principles apply to
patterns in nature. Younger children may notice past and present
seasons; older preschoolers are ready to grasp the cycling of four
seasons in a year.
Teaching strategies. Young children recognize simple patterns on
their own. Complex patterns are more dependent on someone pointing
them out, particularly if the viewer is not looking for them in the
first place. Therefore, teachers can play an especially active role
in helping young children identify and create multi-part repeating
and growing patterns and sequences. For example:
Create complex patterns, then give children art and construction
materials to copy them. Encourage them to create patterns and
series on their own with three or more elements.
Comment on the patterns children create, identifying repeating
elements. For example, Leah showed a painting of two rainbows to
her teacher. It was actually two sequences or patterns of
color that were exactly the same. Look, her teacher commented,
this rainbow has green, red, purple, yellow, and so does this
onegreen, red, purple, yellow (Hohmann & Weikart 2002,
469).
Introduce children to the books and catalogs with complex
patterns used by ceramic tilers, landscape designers (brick and
paver patterns), and fiber artists (weaving, quilting, needlepoint,
basketry). Decorating stores often give away books of discontinued
wallpaper and rug samples. With these, engage children in
describing the patterns and finding corresponding examples that
contain one or more comparable repeated elements in their own
environment; for example, the walkway to the school or a knitted
woolen hat.
Call childrens attention to complex patterns and sequences in
their environment; for example, seasonal cycles, markings on plants
and animals, art and crafts in their community. Encourage children
to duplicate and extend the patterns they see. For example, collect
things with complex patterns on a nature walk and have children
copy and extend the patterns (or create their own comparable one)
at small group or art time.
Provide computer programs that allow children to recognize and
create series and patterns.
Use music to call attention to patterns. Play instrumental music
with pitch, tempo, or loudness patterns and encourage children to
identify them. (This works best if the children are already
familiar with the music.) Sing songs with repeating patterns (where
verses and chorus alternate) or growing patterns (count-down songs
such as I Know an Old Lady Who Swallowed a Fly). Comment on the
patterns and encourage children to identify them.
Use movement to focus on pattern. Older preschoolers can
sequence three movements. If children can master these, encourage
them to be leaders and suggest three-step sequences.
A2 Controlling change Younger children spontaneously notice
changes in themselves and their environment. Older preschoolers not
only observe but can also begin to articulate the reasons for such
changes. Moreover, they can deliberately manipulate variable(s)
to
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produce a desired effect. For example, they may alter the choice
of materials and their arrangement to better represent something in
a collage, or alter the length and angle of a ramp to affect the
speed of a toy car.
Teaching strategies. Teachers can promote awareness of and
curiosity about change by fostering a spirit of inquiry in the
classroom. An adults investigative attitude is transmitted to the
children. They will begin to pose the kinds of questions that
scientists use when they want to know about the properties of
materials and how they operate, then predict and estimate or
measure the results to satisfy their curiosity. Children are eager
to try different things (manipulate variables) and see the
outcomes. Here are some strategies:
Make I wonder what would happen if . . . statements; for
example, I wonder what would happen if you made this end of the
ramp higher.
Ask Suppose you wanted to . . . questions; for example, Suppose
you wanted to make the car go slower. How do you think you could do
that?
During social problem-solving situations (see Chapter 5),
encourage older preschoolers to anticipate the consequences of
their proposed solutions. If they foresee difficulties, have them
consider how to change all or part of the solution to avoid
them.
Data analysis In the preschool years, learning about data
analysis focuses on three elements: Classifying or organizing
involves collecting and categorizing data (e.g., the favorite foods
of children in the class). Representing is diagramming, graphing,
or otherwise recording and displaying the data (e.g., a list of
different foods, with check marks for every child who likes them).
Using information involves asking questions, deciding what data is
needed, and then interpreting the data gathered to answer the
questions (e.g., what to have for snack).
Of the key knowledge and skills in the area of data analysis,
children seem most capable of making collections and
sorting/classifying by attributes
when they learn through child-guided experience and seem most
capable of representing gathered information when they learn
through adult-guided experience.
C Child-guided experience is especially important for learnings
such as: C1 Making collections, sorting/classifying by attributes
Children love to collect and sort things. (Adults do too; science
used to be primarily about collecting specimens and developing
taxonomies to describe each groups characteristics.) Sorting
involves noticing, describing, and comparing the attributes of
things (animals, people, objects) and events. Young children can
classify according to one attribute (e.g., color), and children
slightly older can classify by two attributes (e.g., color and
size). Examples of other attributes by which young children
typically classify include shape, texture, temperature, loudness,
type, and function.
Teaching strategies. Because children are natural collectors,
they will eagerly initiate and respond to suggestions in this area
of mathematical and scientific inquiry. By showing interest in
their collecting and arranging and by asking skillful questions,
teachers can extend child-guided explorations. For example:
Encourage children to make collections of items in the
classroom, natural objects gathered on field trips, and various
objects they bring from home. Provide containers (bowls, boxes,
baskets) for them to sort the items. Encourage them to explain and
describe their collections.
Encourage children to explain why things do not fit into the
categories they have created. For example, pick up an object and
say Would this one fit here?
Provide opportunities to experiment with materials whose
attributes involve all the senses, such as shape, texture, size,
color, pitch, loudness, taste, and aroma.
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Acknowledge and repeat childrens attribute labels, including
invented ones (This fruit feels squishy on my tongue or The pebbles
are bumply). Use common words to build childrens vocabulary (You
used lots of blue in your painting) and introduce new language to
expand their descriptive language (This cloth feels silky or You
used all the rectangular blocks in your tower).
A Adult-guided experience is especially important for learnings
such as: A1 Representing gathered information Representing
information for purposes of data analysis means documenting
categories and quantities with numbers, diagrams, charts, graphs,
counters (e.g., one button for each occurrence), and other symbols.
These activities involve knowledge of both mathematics and
scientific inquiry.
Teaching strategies. Children are naturally curious about their
environment, but their investigations tend to be limited in scope
and haphazard in procedure. Adult intervention can make childrens
explorations and conclusions more systematic and meaningful.
Strategies such as the following help them use the scientific
method to answer questions of interest to them:
Provide materials children can use to record and represent data,
such as clipboards and graph paper.
Pose questions whose answering requires gathering and analyzing
data; for example, How many bags of gerbil food do we need to feed
Pinky for one month? Focus on things of particular interest to
children, such as their bodies (height, age, hair color), animals
and nature (types of pets), the dimensions of things they build,
and what they and their friends like and dislike (foods, favorite
story characters). For example, chart the ingredients children like
best in trail mix, and use the data to make snacks in proportion to
their tastes.
Put a question box in the classroom, and help children write out
and submit questions. For ques
tions that involve data collection, ask children to suggest ways
to answer them.
Be alert to situations that lend themselves to documentation,
such as construction projects that involve multiples of materials.
For example, if children build a train, help them chart the number
of cars or units in the track. If the cars are of different sizes,
create rows or columns and encourage children to record the number
of each. If train building is a recurring activity, investigate
whether trains made on different days are longer or shorter, and by
how many cars.
A2 Interpreting and applying information This component of data
analysis refers to making and testing predictions, drawing
conclusions, and using the results of an investigation to establish
or clarify facts, make plans, or solve problems.
Teaching strategies. Without adult intervention, childrens
scientific inquiries often end with just collecting information.
They may need help to analyze the data to draw one or more
conclusions. Further, childrens learning is less likely to end
there if teachers encourage them to apply their learning to related
topics and to solving problems. Try strategies such as